Question

Write the additive inverse of the complex number of $4-3i$.

Answer 1

Hint: We here have been given a complex number as $4-3i$ and we need to find its additive inverse. Since it is a complex number, its additive inverse will also be a complex number so we will assume its additive inverse to be a complex number given as $x+iy$ and then we will keep the sum of this assumed complex number and the given complex number equal to 0 and then we will solve that equation. Then we will compare the real and imaginary parts on both the sides of the equal sign and hence obtain the values of x and y. Once we have those, we can obtain the value of the required inverse.

Complete step by step solution:
Now, we know that the additive inverse of a number, whether it be complex or real, is the number which when added to the given number gives 0 as the result.
Here, we have been given a complex number as $4-3i$ whose additive inverse we need to find. For this, we first need to know that the inverse of a complex number is always a complex number and that of a real number is always a real number.
Here, since we have been given a complex number, we will assume the inverse of the given complex number to be another complex number given as $x+iy$. Thus, we can say that:
$\left( 4-3i \right)+\left( x+iy \right)=0$
Now, solving it we get:
$\begin{align}
  & \left( 4-3i \right)+\left( x+iy \right)=0 \\
 & \Rightarrow x+iy=-4+3i \\
\end{align}$
Now, comparing the real and the imaginary parts of this equation, we get:
$\begin{align}
  & x=-4 \\
 & y=3 \\
\end{align}$
Thus, putting the values of x and y we get:
$\begin{align}
  & x+iy \\
 & \therefore -4+3i \\
\end{align}$
Hence, the additive inverse of 4-3i is -4+3i.

Note: We need not assume the complex number in the form of $x+iy$ to solve this question. We can also straightaway assume the inverse to be a complex number ‘z’ and form the equation and directly get the value of z. This is shown as follows:
$\begin{align}
  & \left( 4-3i \right)+z=0 \\
 & \therefore z=-4+3i \\
\end{align}$
Also, just like real numbers, the additive inverse of a complex number is just the negation of the given complex number.